Stefan Problems with Supercooling: Theory and Applications

Abstract

We consider certain McKean-Vlasov stochastic differential equations (SDEs) which arise as mean-field limits of particle systems with interactions through the hitting times of free boundaries. The interaction term in these equations is self-exciting, which can lead to short-time blowup in the form of jump discontinuities. Recently, such equations have found use as probabilistic representations for solutions of the supercooled Stefan problem, a free boundary partial differential equation (PDE), which describes the freezing of supercooled liquids; and they have also been used to model systemic risk in interconnected financial networks. In this work, we show existence of physical solutions for a two-phase version of this model: any possible discontinuities of these physical solutions must obey a natural physicality condition for the problem at hand. Additionally, we investigate the well-posedness of the one-phase problem by considering the sensitivity to perturbed boundary and initial conditions for physical solutions, as well as minimal solutions, another solution concept.